Let $A$ be a subset of $\mathbb{R}^n$ then the rearrangement of $A$ denoted by $A^*$ is the ball $B(0,r)$ having the same volume as $A$ i.e if $|A| =|B(0,r)|$ with respect to Lebesgue measure then $$A^*= B(0,r)$$ Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}$. Then its symmetric de...

I'm studying section 3.3 of Analysis by Lieb and Loss, about symmetric-decreasing rearrangement of functions. Let $A\subset \mathbb{R}^n$ a Borel set of finite Lebesgue measure. They define $A^*$ to be the ball centered at 0 with the same measure that $A$. The symmetric-decreasing re...

I found this statement about rearrangement from analysis Lieb and Loss in chapter 3. Suppose f, g are nonnegative functions in $L^2(\Bbb{R^n})$, then $||f^*-g^*||_2 \le||f-g||_2$ Where $f^*$ is the symmetric- decreasing rearrangement of $f$. $f^*(x):=\int_0^{\infty} \chi_{\{|f|>t\}^*}(x)dt$. ...

Don't know how many of you guys are familiar with the theory of rearrangements, but I have a question for you about it. As you can see in Leoni (or in Lieb & Loss), the decreasing and the increasing rearrangement of a measurable function $u:\Omega \to [0,\infty[$ ($\Omega \subseteq \mathbb{R}^N...

Let $f \, \colon \mathbb{R}^d \rightarrow[0, \infty)$ be a function such that the sets $ \{ y \: \colon f(y) > \lambda \}$ are of finite Lebesgue-measure, for every $\lambda \geq 0$. Then, we can define the spherically symmetric and decreasing rearrangement $f^* \, \colon \mathbb{R}^d \rightarrow...

Let $(X,dx)$ a measure space and $f\in L^p(X,\mathbb{C})$; let's define its distribution function $$F(\alpha)=meas(\{x\in X||f(x)|>\alpha\})$$ and the decreasing rearrangement $$\alpha_k=\inf\{\alpha>0|F(\alpha)<2^k\}$$ I have to prove the following $$\sum_{k:\alpha_k>\alpha}2^k\leq 2 F(\alpha)$$...

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: u(x)\geq t \} > \omega_N |x|^N\} $$ where $\mu$ is the Lebesgue measure in $\mathbb{R}^N$, $w_N=\...

If $f(x)=|\arctan(x)|$, how does one constructively prove that its decreasing rearrangement is given by the constant function $f^\star(y)=\pi/2$ defined on $[0,\infty)$? The decreasing rearrangement of a function $f$ is defined as $$f^\star(x)=\inf\{u:|\{t:f(t)>y\}|\leq x\}$$

Two functions $f(x)$ and $g(x)$ are called equimeasurable if $m(\{x:f(x)>t\})=m(\{x:g(x)>t\})$. Nondecreasing rearrangement of a function $f(x)$ is defined as $$f^*(\tau)=\inf\{t\in \mathbb{R}^1:m(\{x:f(x)>t\}\leq\tau\}.$$ Prove that $f^*(\tau)$ and $f(x)$ are equimeasurable.